We analyze the Gr{\o}stl hash function, which is a 2nd-round candidate of the SHA-3 competition.
Using the start-from-the-middle variant of the rebound technique, we
show collision attacks on the Gr{\o}stl-256 hash function reduced to
5 and 6 out of 10 rounds with time complexities $2^{48}$ and $2^{112}$, respectively.
Furthermore, we demonstrate semi-free-start collision attacks on the
Gr{\o}stl-224 and -256 hash functions reduced to 7 rounds and the Gr{\o}stl-224 and -256 compression functions reduced to 8 rounds.
Our attacks are based on differential paths between the two permutations $P$ and $Q$ of Gr{\o}stl, a strategy introduced by Peyrin to construct distinguishers for the compression function.
In this paper, we extend this approach to construct collision and semi-free-start collision attacks for both the hash and the compression function.
Finally, we present improved distinguishers for reduced-round versions of the Gr{\o}stl-224 and -256 permutations.